mixedradix

Mixed radix is a numeral system in which the bases of successive digit positions vary. A mixed radix representation uses a sequence of bases r0, r1, r2, ... and digits a0, a1, a2, ..., with 0 ≤ a_i < r_i. The value of the representation is a0 + a1*r0 + a2*(r0*r1) + a3*(r0*r1*r2) + ... . In other words, each position carries a digit that multiplies the product of all earlier bases.

Example: with bases 60, 60, and 24 (as in seconds, minutes, and hours), a number is written as (a0, a1, a2) with a0 in 0–59, a1 in 0–59, and a2 in 0–23. To encode 123 seconds, a0 = 123 mod 60 = 3; q1 = floor(123/60) = 2; a1 = q1 mod 60 = 2; a2 = floor(q1/60) = 0. The digits (3, 2, 0) correspond to 0 hours, 2 minutes, and 3 seconds, since 3 + 2*60 + 0*3600 = 123.

Relation to other systems: The factorial number system is a common mixed radix where bases are 2, 3, 4, 5, ... and digits satisfy 0 ≤ a_i ≤ i. This system underlies permutation indexing and certain combinatorial encodings.

Applications and notes: Mixed radix is used in timekeeping, calendars, measurement systems with non-uniform bases, digital arithmetic, and encoding multi-dimensional indices. It provides a flexible alternative to fixed-base representations and is central to both theoretical and practical numbering schemes.